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Instantaneous frequency estimation of FMsignals by CB-energy operator

A.-O. Boudraa

The CB energy operator is an extension of the cross-Teager-Kaiser

energy operator which is a nonlinear energy tracking operator to deal

with complex signals and its usefulness for non-stationary signals

analysis has been demonstrated. Two new properties ofCB are estab-

lished. The first property is the link betweenCB and the dynamic signal

which is a generalisation of the instantaneous frequency (IF). The

second property obtained for frequency modulated (FM) signals is a

simple way to estimate the IF. These properties confirm the interest

of the CB operator to track the non-stationarity of a signal. Results

of IF estimation in a noisy environment of a nonlinear FM signal are

presented and comparison to the Wigner-Ville distribution and the

Hilbert transform-based method is provided.

Introduction: The CB-energy operator is an extension of the cross-

Teager-Kaiser operator which is a nonlinear tracking energy operator

[1] to deal with complex signals [2]. Based on the Lie bracket, CB is

well suited to measure the instantaneous interaction between two

complex signals [3]. The output ofCB is related to the instantaneous

cross-correlation (CC) of complex signals x(t) and y(t), Rxy(t, t), as

follows [2]:

CB(x(t),y(t)) = 2Rxy(t, t)

t2

t=0

2Rxy(t, t)t2

t=0

(1)

Relation (1) shows thatCB is a cross-energy function of two signals and

thus links to transforms using the concept of instantaneous CC, such as

time-frequency representations (TFRs), can be found. In fact, relation-

ships between theCB operator and some TFRs such as cross-ambiguity

function or cross-Wigner-Ville Distribution (WVD) show thatCB is

well suited to study non-stationary signals [24]. CB has found appli-

cations in both signal and image processing such as time series analysis,

gene time series expression data clustering, transient detection or time

delay estimation [58]. The CB operator is a symmetric bilinear form

defined as follows [2]:

CB

(x,y

) =0.5

[CC

(x,y

) +CC

(y,x

)] (2

)where the associated quadratic form CC(x, x) CB(x, x) is given by

CC(x,y) = 0.5[xy + xy] 0.5[xy + xy] (3)

In this Letter new properties ofCB are established. We show the link

between CB and the dynamic signal which describes the rate of the

log-magnitude and phase [9]. The second relationship is a simple way

to estimate the IF of a frequency modulated (FM) signal. Thus, it is

natural to use CB as the local operator, based of the signal time deriva-

tives, as a basis for tracking instantaneous features such as the instan-

taneous frequency (IF) function. In the following, for x(t) y(t) the

notation CB(x(t), y(t)) ; CB(x(t)) is used.

Dynamic signal andCB operator: The dynamic signal is similar to the

definition of complex cepstrum in the frequency domain and can beviewed as a generalisation of the IF of a signal with varying magnitude

[9]. Consider an AM-FM signal

x(t) = a(t)ejf(t) (4)

a(t) is instantaneous amplitude andf(t) instantaneous phase. The IF is

given by

fi(t) =1

2pf(t) (5)

The dynamic signal is defined by [9]

b(t) = ddt

logx(t) = a(t)a(t) + jf(t) (6)

and the instantaneous bandwidth is given by

ib(t) = a(t)a(t)

(7)

If one substitutes x(t) (4) in (2), then one obtains

CB(x(t)) = |x(t)|2

2(b(t) b(t))2 + (b(t) + b(t)) (8)

which shows thatCB can be written in terms of a dynamic signal.

Furthermore, (8) can also be written as

CB(x(t)) = |x(t)|2 8p2f2i (t) + ib2(t) a(t)a(t)

(9)

where there is a relationship betweenC

B, the IF and the instantaneousbandwidth. Finally, links given by relations (8) and (9) show thatCBconveys precious local information about the instantaneous behaviour,

over the time, of a signal.

For a FM signal (a(t) A), using (9) the IF is given by

fi(t) =CB(x(t))2pA

2

(10)

where A is a constant. Relation (10) is a simple way to estimate the IF of

a FM signal. This IF estimation is obtained without involving integral

transform and with no a priori knowledge about the phase f(t) of the

signal. An advantage of the CB operator is its localisation property.

This localisation is the consequence of the differentiation based method.

Results: Consider a noisy polynomial phase signal (PPS):

y(t) = z(t) + n(t) = Aejf(t) + n(t)

= AejPk=0

aktk

+ n(t), t[ [0, T](11)

where z(t) is a noise-free signal, n(t) is a complex white Gaussian noise,

f(t) is signal phase, P is order of polynomial and ak are corresponding

polynomial coefficients. T is signal duration. A fourth-order PPS, with

parameters A 2 and (a0 1, a1 10, a2 10, a3 2, a4 25),

is selected and Monte-Carlo simulations implemented to show the effec-

tiveness ofCB (10) as an IF estimator. Estimation is evaluated for an

input signal-to-noise ratio (SNR) ranging from 26 to 40 dB with an

interval of 2 dB. One hundred realisations are performed for each

SNR and mean square error (MSE) between the estimated value anddesired value chosen as performance criteria. Since both CB (10) and

the unwrapped phase of the analytical signal (Hilbert transform (HT))

(5) involve derivatives, numerical differentiations are used. For very

noisy signals, traditional Euler derivatives are useless. A better option

is to use derivatives filters such as the Savitzky-Golay (SG) differen-

tiation filter [10]. This filter uses a polynomial fit across a moving

window that preserves higher-order moments of the signal.

Derivatives are calculated with a fourth-order SG filter and 33 point

windows. In Fig. 1a we display the IF (ideal) of z(t). Figs. 1bd

show the results of IF tracking using HT, WVD and CB-based

methods for SNR 30 dB. Compared to HT and VWD, the best extrac-

tion is given by the CB approach. Fig. 1b shows an agreement of VWD

estimate with true IF but with fluctuations of high magnitude due essen-

tially to cross-terms corruption in the presence of noise [11]. HT shows a

good match but with ripples (modulations errors) (Fig. 1c) compared toCB (Fig. 1d).

Fig. 2 shows MSE against input SNR for the HT, WVD andCB-based

methods. This result shows the ability ofCB to resolve a nonlinear FM

component. As seen in Fig. 2, apart from different SNRs, the CBapproach provides a significant performance improvement over the HT

and VWD approaches. Of particular interest are the results at low

SNRs for which the WVD approach performs less well than CB and

HT. This is expected, as pointed out in [12]. As the SNR decreases,

WVD produces biased representation of signal energy as well as spur-

ious components (artefacts). Except for higher SNRs (30 dB), HToutperforms the WVD approach mainly due to use of an integral transform

which does implicit smoothing (lowpass filtering) [11]. For SNR36 dB, CB and DWV have similar performance. It is important to

mention thatCB, like most local approaches or those based on signal

derivatives, is sensitive to a very noisy environment. For moderatelynoisy data, classical Euler discretisation schemes are robust to noise.

For low SNRs, the use of derivatives filters such as SG is necessary

for efficient tracking of the IF by CB or HT. It was found that for differ-

ent polynomial orders 3 the obtained results are similar but the

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precision of estimates both by CB and HT are dependent on the size of

the SG sliding window used.

8

10

12

14

16

18

20

22

frequency,

Hz

true IF

a

WVD

b

0.2 0.4 0.6 0.8 18

10

12

14

16

18

20

22

time, s

frequency,

Hz

HT

c

0.2 0.4 0.6 0.8 1

time, s

PsiBSG

d

Fig. 1 IF estimate for fourth-order PPS

a Trueb WVDc HT

dCB

5 0 5 10 15 20 25 30 35 40

10

0

10

20

30

40

50

60

70

80

90

SNR, dB

MSE,

dB

Psi_BWVDHT

Fig. 2 MSE in polynomial IF estimation for HT, WVD, CB operator

Conclusion: In this Letter, two new properties of the CB energy oper-

ator are presented. The link between CB and the dynamic signal

shows that CB conveys precious local information about the instan-

taneous behaviour, over time, of a signal. The second property obtained

for FM signals is a simple way to estimate the IF with no a priori

knowledge about the phase of the signal. Preliminary results show

thatCB is effective for estimating the IF of a nonlinear FM signal com-

pared to HT and WVD based approaches. Further, the computational

cost ofCB is much lower. As future work we plan to apply CB to a

wide range of synthetic and real signals to confirm the obtained

results and to explore new properties ofCB.

# The Institution of Engineering and Technology 2011

4 March 2011

doi: 10.1049/el.2011.0586

One or more of the Figures in this Letter are available in colour online.A.-O. Boudraa ( Ecole Navale, IRENav, BCRM Brest CC 600, Brest

Cedex 9 29240, France)

E-mail: boudra@ecole.navale.fr

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